Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5

Average Error: 20.2 → 4.0

Time: 5.7s

Precision: binary64

\[ \begin{array}{c}[x, y, z] = \mathsf{sort}([x, y, z])\\ \end{array} \]

Math

\[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

↓

\[\begin{array}{l} t_0 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ t_1 := 2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-204}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+40}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (* 2.0 (pow (exp (* 0.25 (- (log (+ y x)) (log (/ 1.0 z))))) 2.0)))
        (t_1
         (*
          2.0
          (pow (exp (* 0.25 (- (log (- (- z) y)) (log (/ -1.0 x))))) 2.0))))
   (if (<= y -4.6e+54)
     t_1
     (if (<= y -5e-204)
       (* 2.0 (sqrt (* x (+ y z))))
       (if (<= y -1.65e-303)
         t_1
         (if (<= y 1.16e-183)
           t_0
           (if (<= y 1e+40) (* 2.0 (sqrt (* z (+ y x)))) t_0)))))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
}

↓

double code(double x, double y, double z) {
	double t_0 = 2.0 * pow(exp((0.25 * (log((y + x)) - log((1.0 / z))))), 2.0);
	double t_1 = 2.0 * pow(exp((0.25 * (log((-z - y)) - log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -4.6e+54) {
		tmp = t_1;
	} else if (y <= -5e-204) {
		tmp = 2.0 * sqrt((x * (y + z)));
	} else if (y <= -1.65e-303) {
		tmp = t_1;
	} else if (y <= 1.16e-183) {
		tmp = t_0;
	} else if (y <= 1e+40) {
		tmp = 2.0 * sqrt((z * (y + x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 * (exp((0.25d0 * (log((y + x)) - log((1.0d0 / z))))) ** 2.0d0)
    t_1 = 2.0d0 * (exp((0.25d0 * (log((-z - y)) - log(((-1.0d0) / x))))) ** 2.0d0)
    if (y <= (-4.6d+54)) then
        tmp = t_1
    else if (y <= (-5d-204)) then
        tmp = 2.0d0 * sqrt((x * (y + z)))
    else if (y <= (-1.65d-303)) then
        tmp = t_1
    else if (y <= 1.16d-183) then
        tmp = t_0
    else if (y <= 1d+40) then
        tmp = 2.0d0 * sqrt((z * (y + x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((y + x)) - Math.log((1.0 / z))))), 2.0);
	double t_1 = 2.0 * Math.pow(Math.exp((0.25 * (Math.log((-z - y)) - Math.log((-1.0 / x))))), 2.0);
	double tmp;
	if (y <= -4.6e+54) {
		tmp = t_1;
	} else if (y <= -5e-204) {
		tmp = 2.0 * Math.sqrt((x * (y + z)));
	} else if (y <= -1.65e-303) {
		tmp = t_1;
	} else if (y <= 1.16e-183) {
		tmp = t_0;
	} else if (y <= 1e+40) {
		tmp = 2.0 * Math.sqrt((z * (y + x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

↓

def code(x, y, z):
	t_0 = 2.0 * math.pow(math.exp((0.25 * (math.log((y + x)) - math.log((1.0 / z))))), 2.0)
	t_1 = 2.0 * math.pow(math.exp((0.25 * (math.log((-z - y)) - math.log((-1.0 / x))))), 2.0)
	tmp = 0
	if y <= -4.6e+54:
		tmp = t_1
	elif y <= -5e-204:
		tmp = 2.0 * math.sqrt((x * (y + z)))
	elif y <= -1.65e-303:
		tmp = t_1
	elif y <= 1.16e-183:
		tmp = t_0
	elif y <= 1e+40:
		tmp = 2.0 * math.sqrt((z * (y + x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(Float64(Float64(x * y) + Float64(x * z)) + Float64(y * z))))
end

↓

function code(x, y, z)
	t_0 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(y + x)) - log(Float64(1.0 / z))))) ^ 2.0))
	t_1 = Float64(2.0 * (exp(Float64(0.25 * Float64(log(Float64(Float64(-z) - y)) - log(Float64(-1.0 / x))))) ^ 2.0))
	tmp = 0.0
	if (y <= -4.6e+54)
		tmp = t_1;
	elseif (y <= -5e-204)
		tmp = Float64(2.0 * sqrt(Float64(x * Float64(y + z))));
	elseif (y <= -1.65e-303)
		tmp = t_1;
	elseif (y <= 1.16e-183)
		tmp = t_0;
	elseif (y <= 1e+40)
		tmp = Float64(2.0 * sqrt(Float64(z * Float64(y + x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((((x * y) + (x * z)) + (y * z)));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = 2.0 * (exp((0.25 * (log((y + x)) - log((1.0 / z))))) ^ 2.0);
	t_1 = 2.0 * (exp((0.25 * (log((-z - y)) - log((-1.0 / x))))) ^ 2.0);
	tmp = 0.0;
	if (y <= -4.6e+54)
		tmp = t_1;
	elseif (y <= -5e-204)
		tmp = 2.0 * sqrt((x * (y + z)));
	elseif (y <= -1.65e-303)
		tmp = t_1;
	elseif (y <= 1.16e-183)
		tmp = t_0;
	elseif (y <= 1e+40)
		tmp = 2.0 * sqrt((z * (y + x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[(N[(x * y), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(1.0 / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Power[N[Exp[N[(0.25 * N[(N[Log[N[((-z) - y), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+54], t$95$1, If[LessEqual[y, -5e-204], N[(2.0 * N[Sqrt[N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.65e-303], t$95$1, If[LessEqual[y, 1.16e-183], t$95$0, If[LessEqual[y, 1e+40], N[(2.0 * N[Sqrt[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	20.2
Target	11.4
Herbie	4.0

\[\begin{array}{l} \mathbf{if}\;z < 7.636950090573675 \cdot 10^{+176}:\\ \;\;\;\;2 \cdot \sqrt{\left(x + y\right) \cdot z + x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right) \cdot \left(0.25 \cdot \left(\left({y}^{-0.75} \cdot \left({z}^{-0.75} \cdot x\right)\right) \cdot \left(y + z\right)\right) + {z}^{0.25} \cdot {y}^{0.25}\right)\right) \cdot 2\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if y < -4.59999999999999988e54 or -5.0000000000000002e-204 < y < -1.6499999999999999e-303

   1. Initial program 41.2

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Simplified 41.2

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]

   3. Applied egg-rr 41.3

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in x around -inf 6.6

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{-1}{x}\right) + \log \left(-1 \cdot z + -1 \cdot y\right)\right)}\right)}}^{2} \]

   if -4.59999999999999988e54 < y < -5.0000000000000002e-204

   1. Initial program 2.0

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Simplified 2.0

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]

   3. Taylor expanded in x around inf 2.0

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(y + z\right) \cdot x}} \]

   if -1.6499999999999999e-303 < y < 1.16000000000000006e-183 or 1.00000000000000003e40 < y

   1. Initial program 39.2

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Simplified 39.2

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]

   3. Applied egg-rr 39.3

      \[\leadsto 2 \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)\right)}^{0.25}\right)}^{2}} \]

   4. Taylor expanded in z around inf 6.5

      \[\leadsto 2 \cdot {\color{blue}{\left(e^{0.25 \cdot \left(-1 \cdot \log \left(\frac{1}{z}\right) + \log \left(y + x\right)\right)}\right)}}^{2} \]

   if 1.16000000000000006e-183 < y < 1.00000000000000003e40

   1. Initial program 1.2

      \[2 \cdot \sqrt{\left(x \cdot y + x \cdot z\right) + y \cdot z} \]

   2. Simplified 1.2

      \[\leadsto \color{blue}{2 \cdot \sqrt{\mathsf{fma}\left(x, y, z \cdot \left(x + y\right)\right)}} \]

   3. Taylor expanded in z around inf 1.2

      \[\leadsto 2 \cdot \sqrt{\color{blue}{\left(y + x\right) \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+54}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-204}:\\ \;\;\;\;2 \cdot \sqrt{x \cdot \left(y + z\right)}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-303}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(\left(-z\right) - y\right) - \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-183}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \mathbf{elif}\;y \leq 10^{+40}:\\ \;\;\;\;2 \cdot \sqrt{z \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {\left(e^{0.25 \cdot \left(\log \left(y + x\right) - \log \left(\frac{1}{z}\right)\right)}\right)}^{2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64

  :herbie-target
  (if (< z 7.636950090573675e+176) (* 2.0 (sqrt (+ (* (+ x y) z) (* x y)))) (* (* (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25))) (+ (* 0.25 (* (* (pow y -0.75) (* (pow z -0.75) x)) (+ y z))) (* (pow z 0.25) (pow y 0.25)))) 2.0))

  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))